Anders Lee and Simon Holmstrom for the Islanders (21-13-4), who have points in three straight (2-0-1) and face the Columbus Blue Jackets on Sunday for the second of a back-to-back.
The Rangers (19-17-4) were shut out for the eighth time this season. Igor Shesterkin made 24 saves.
Lee gave the Islanders a 1-0 lead at the 58-second mark of the first period. Rangers defenseman Carson Soucy turned the puck over in front to forward Mathew Barzal. Shesterkin made the initial save on Barzal, but Lee beat him blocker side on the rebound to extend his personal point streak to five games.
Rittich slid across his crease to deny a Will Cuylle 2-on-1 try at the back post at 11:26. The Islanders were up 5-0 in shots until that point.
Cuylle thought he scored to tie it 1-1 at 14:37 of the second period on the power play. After initially being called a goal on the ice, it was determined that the puck never crossed the goal line and his wrist shot instead went off the left post.
Rittich denied Soucy’s penalty shot with the left pad at 9:18 of the third period to keep the Islanders in front. Forward Calum Ritchie drew the penalty on Soucy with a slash on the breakaway attempt.
Holmstrom scored on the empty-net goal at 19:43 of the third period for the 2-0 final.